Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space

摘要

Abstract In this paper, we successfully give two interesting lower bounds for the first eigenvalue of submanifolds (with bounded mean curvature) in a hyperbolic space. More precisely, let M be an n-dimensional complete noncompact submanifold in a hyperbolic space and the norm of its mean curvature vector ‖ H ‖ satisfies ‖ H ‖ ? α n ? 1 , then we prove that the first eigenvalue λ 1 , p ( M ) of the p-Laplacian Δ p on M satisfies λ 1 , p ( M ) ? ( n ? 1 ? α p ) p